In many virus families, replication requires that hundreds to thousands of proteins assemble around the viral nucleic acid (NA) to form a protein shell called a capsid. Furthermore, many animal viruses use protein assembly to drive budding of the capsid from a cell membrane. Understanding the mechanisms that control assembly around NAs and on membranes would identify targets for novel antivirus therapies that inhibit NA packaging or budding, and would guide efforts to exploit viruses as targeted transport vehicles. Assembly mechanisms inferred from experiments alone are incomplete because intermediates are transient. Therefore, this project develops and applies computational models for capsid proteins, NAs, and lipids that reveal details of assembly and membrane budding not accessible to experiments. To understand how the properties of viral NAs facilitate assembly, models are developed for capsid proteins and NAs that begin with a linear polyelectrolyte (without base-pairing) and then systematically add the geometric and electrostatic features of NAs that arise due to base-pairing. Comparison of predicted assembly kinetics and thermodynamics for each model identifies the contributions of base-pairing to assembly. Predictions for each model are tested against experiments performed by collaborating labs on capsid assembly around corresponding molecules (e.g., synthetic polyelectrolytes, heterologous NAs, and viral genomic NAs). The mechanism by which capsids form different icosahedral morphologies to accommodate NAs with different sizes is also studied. Employed simulation techniques include Brownian dynamics and equilibrium calculations. For some enveloped viruses (e.g., HIV) capsid assembly drives budding from a cell membrane, while for others (e.g., alphaviruses) assembly of membrane proteins drives budding of a pre-assembled capsid. Simulations are used to investigate how these two classes of assembly-driven budding processes depend on properties such as protein interactions and membrane rigidity, and why many viruses preferentially bud from particular membrane microdomains. Predictions will be compared to experiments on alphavirus budding. In addition to identifying factors that can be manipulated to prevent or exploit viral assembly, the proposed simulations will elucidate how biology employs membranes and filamentous scaffolds to assemble multi- macromolecular complexes. The research combines coarse-grained models that are informed by atomistic simulations and experiments with recent advances in GPUs and distributed computing to simulate relevant time and length scales. A new method to apply Markov state models to assembly reactions is developed.